Chapter 1 An Overview Of Weibull Analysis 
11 
1.1 Objective 
11 
1.2 Background 
11 
1.3 Examples 
12 
1.4 Scope 
12 
1.5 Advantages of Weibull Analysis 
13 
1.6 Data, Discrete Versus Life Data 
13 
1.7 Failure Distribution 
14 
1.8 Failure Forecasts And Predictions 
15 
1.9 Engineering Change Test Substantiation 
16 
1.10 Maintenance Planning 
16 
1.11 System Analysis And Math Models 
17 
1.12 Weibulls With Curved Data 
17 
1.13 Weibulls With Corners And Doglegs 
19 
1.14 Weibayes 
19 
1.15 Small Sample Weibulls 
19 
1.16 Updating Weibulls 
19 
1.17 Deficient (Dirty) Data 
19 
1.18 Establishing The Weibull Line, Choosing The Fit Method 
110 
1.19 Related Methods And Problems 
110 
1.20 Summary 
111 
Chapter 2 Plotting The Data And Interpreting The Plot 
21 
2.1 Foreword 
21 
2.2 Weibull Data 
21 
2.3 The Weibull Plot Scales 
22 
2.4 (Eta) and (Beta) 
22 
2.5 Weibull Analysis – An Example 
24 
2.6 Median Ranks 
25 
2.7 The Weibull Plot 
26 
2.8 “B” Life 
26 
2.9 Suspended Test Items 
27 
2.10 Bernard’s Approximation 
27 
2.11 Suspensions Increase Eta 
28 
2.12 Interpreting The Weibull Plot 
28 
2.13 Beta < 1 Implies Infant Mortality 
29 
2.14 Beta = 10 Implies Random Failures 
210 
2.15 10 < Beta < 40 Implies Early Wear Out 
211 
2.16 Beta > 40 Implies Old Age (Rapid) Wear Out 
211 
2.17 Weibull Modes May Be “Covered” 
212 
2.18 Weibull Paper And Its Construction 
212 
2.19 Weibull Analysis – The Standard Method 
214 
2.20 Problems 
213 
Chapter 3 Dirty Data, “Bad” Weibulls, And Uncertainties 
31 
3.1 Foreword 
31 
3.2 Small Sample Uncertainties 
31 
3.2.1 Goodness Of Fit 
33 
3.3 Suspensions 
36 
3.4 Suspect Outliers 
36 
3.5 Curved Weibulls And The t0 Correction 
37 
3.6 Curved Weibulls And The Log Normal Distribution 
311 
3.7 Data Inconsistencies And Multimode Failures 
314 
3.7.1 LowTime Failures 
314 
3.7.2 Close Serial Numbers 
315 
3.7.3 Mixtures Of Failure Modes 
316 
3.8 Steep Slopes Hide Problems 
317 
3.9 Bad Weibull Patterns 
318 
Conclusion 
318 
3.10 Problems 
319 
Chapter 4 Failure Forecasting = Risk Analysis 
41 
4.1 Situation 
41 
4.2 Definition 
41 
4.3 Forecasting Techniques 
41 
4.4 Calculating Failure Forecasts 
41 
4.4.1 Expected Failures Now 
41 
4.1.2 Failure Forecast When Failed Units Are Not Replaced 
43 
4.4.3 Failure Forecasts When Failed Units Are Replaced 
43 
4.5 Failure Forecast AnalysisSummary 
44 
4.5.1 Case Study 1: Bearing Cage Fracture 
45 
4.5.2 Case Study 2: Bleed System Failures 
47 
4.6 System Failure Forecast Without Simulation* 
412 
4.6.1 Case Study 3: Aircraft InFlight Engine Shutdowns* 
412 
4.7 System Failure Forecasts With Simulation* 
415 
4.7.1 Case Study 4: System Failure Forecast With Simulation* 
417 
4.8 Optimal (Lowest Cost) And Block Replacement Intervals* 
419 
4.9 Problems 
425 
Chapter 5 Maximum Likelihood Estimates & Other Alternatives 
51 
5.1 Introduction 
51 
5.2 Maximum Likelihood Estimation (MLE) 
51 
5.3 MLE With Reduced Bias Adjustment (RBA) for Accurate Results 
53 
5.3.1 The RBA Factor for Normal and Lognormal Distributions 
54 
5.3.2 The RBA factor for the Weibull distribution 
55 
5.3.3 Best Practice 
56 
5.4 Median Rank Regression: X on Y Versus Y on X 
57 
5.5 Plotting Positions 
59 
5.5 Special Methods: MLE With Reduced Bias Adjustment (RBA) 
56 
5.6 Special Methods: Gossett’s Student’s T 
510 
5.7 The Dauser Shift – Unknown Suspension Times 
510 
5.8 Special Methods For Inspection Interval Data And Coarse Data 
512 
5.8.1 Inspection Option #1 
512 
5.8.2 & 

5.8.3 Probit Analysis Inspection Options #2 & 3 
513 
5.8.4 KaplanMeier (KM) Inspection Option #4 
514 
5.8.5 Interval Maximum Likelihood Estimation (MLE) Inspection Option #5 
514 
5.9 Distribution Analysis 
516 
Chapter 6 Weibayes And Weibayes Substantiation Testing 
61 
6.1 Foreword 
61 
6.2 Weibayes Method 
62 
6.3 Weibayes Without Failures 
62 
6.4 Weibayes With Failures 
63 
6.5 Unknown Failure Times 
64 
6.6 Weibayes Worries And Concerns 
64 
6.7 Weibayes Case Studies 
65 
6.8 Substantiation And Reliability Testing 
69 
6.9 ZeroFailure Test Plans For Substantiation Testing 
610 
6.10 ZeroFailure Test Plans For Reliability Testing 
612 
6.10.1 ReExpression Of Reliability Goal To Determine ? 
612 
6.10.2 Tailoring and Designing Test Plans 
614 
6.11 Total Test Time 
615 
6.12 TestToFailure Versus Weibayes ZeroFailure Tests 
616 
6.13 One Or Zero Failure Test Plans 
619 
6.14 Sudden Death Tests With Weibull And Weibayes 
620 
6.15 Case Study: Cost Vs Uncertainty Trades 
623 
6.16 Normal And Lognormal Tests 
622 
6.17 Accelerated Testing 
624 
6.17.1 Accelerated StepStress Test Data Analysis* 
625 
6.17.2 Accelerated Testing: A Method For Estimating Test Acceleration Factor With No Existing
InService Failures* 
626 
6.18 System Deterioration 
628 
6.19 Weibull Libraries And Lessons Learned 
629 
6.19.1 A Weibull Library Case Study 
630 
6.19.2 Weibull Libraries For End Users 
631 
6.21 Problems 
632 
Chapter 7 Interval Estimates 
71 
7.1 Interval Estimates 
71 
7.2 Confidence Interval Concept 
71 
7.3 Confidence Intervals For B Lives And Reliability 
72 
7.3.1 BetaBinomial Bounds 
73 
7.3.2 Fisher’s Matrix Bounds 
74 
7.3.3 Likelihood Ratio Bounds 
75 
7.3.4 Pivotal Bounds Monte Carlo Bounds 
76 
7.3.5 Reliability Assurance Interval and the “p” value 
77 
7.3.6 Normal Distribution Confidence Bounds with Student’s t 
77 
7.3.7 Summary Of Confidence Bounds For B Life And Reliability 
78 
7.4 Confidence Intervals For Eta And Beta 
78 
7.5 Are Two Weibull Data Sets Different Or From The Same Distribution 
79 
7.5.1 Double Confidence Bounds Do Not Overlap 
710 
7.5.2 Likelihood Ratio Test 
711 
7.5.3 Likelihood Contour Plots 
711 
7.6 Problems – True Or False? 
713 
Chapter 8 Related Math Models 
81 
8.1 Introduction 
81 
8.2 Binomial Distribution 
81 
8.3 Poisson Distribution 
85 
8.4 Binomial Becomes Poisson…Sometimes 
89 
8.5 The Exponential Distribution 
811 
8.6 KaplanMeier Survival Estimates 
812 
8.7 Probabilistic Design 
817 
8.7.1 StrengthLoad And LifeUsage Interactions 
817 
8.7.2 Total Life = Crack Life + CrackToRupture Life 
818 
8.7.3 Does Failure Mode A Cover Mode B? 
819 
8.8 Production Process Reliability 
819 
8.9 Extreme Value Statistics 
821 
8.10 Batch Effects 
823 
8.11 Problems 
824 
Chapter 9 CrowAMSAA Modeling, Warranty Analysis & Life Cycle Costs 
91 
9.0 The CrowAMSAADuane Reliability Growth Model 
91 
9.1 Background History 
92 
9.2 CA Methods 
92 
9.2.1 Simple Graphical and Regression Solution 
92 
9.2.2 IEC Solutions for Time and Failure Terminated Data 
95 
9.2.3 IEC MLE Solutions for Interval and Grouped Data 
97 
9.3 Comparisons of the IEC and Regression CA Methods 
912 
9.4 CA Input May Be Confusing 
914 
9.5 Missing Early Data with CA 
914 
9.6 First Time Failure Mode Analysis 
914 
9.7 Warranty Claims Analysis 
915 
9.8 Warranty Data Matrix 
916 
9.9 Warranty Data Rules 
917 
9.10 Warranty Data Matrix Conversion and Analysis 
918 
9.11 Warranty Analysis Methods 
920 
9.11.1 Inspection Option #1 
920 
9.11.2 KaplanMeier 
920 
9.11.3 MLE Interval 
920 
9.11.4 Crow AMSAA 
920 
9.12 Case Studies 
921 
9.13 Tracking Your Results 
921 
9.14 Warranty Conclusions and Recommendations 
921 
9.15 Life Cycle Cost 
921 
9.16 Net Present Value (NPV) 
921 
9.17 Discount Rate 
922 
9.18 Life Cycle Cost and NPV 
922 
9.19 LCC Calculations 
923 
9.20 Case Studies 
924 
Chapter 10 Summary 
101 
10.1 The Beginning Of The End 
101 
10.2 Which Method? What Kind Of Data? 
101 
10.3 Looking At The Plot, What Do You See? 
103 
10.4 Which Distribution Is Best? 
104 
10.5 Substantiation And Accelerated Testing 
106 
10.6 Confidence Intervals 
106 
10.7 Presentations And Reports 
106 
10.8 Logic Diagram – Flowchart 
106 
10.9 The End 
106 
10.10 Best Practice Flow Chart 
107 
Chapter 11 Case Studies And New Applications 
111 
11.1 Foreword 
111 
11.2 Stress Corrosion Failure Forecasting 
112 
11.3 Optimal Component Replacement – Voltage Regulators 
113 
11.4 Locomotive Power Units Overhaul Life 
117 
11.5 Cost Effective Calibration Intervals 
118 
11.6 Florida Power & Light Turbogenerator Failure 
1110 
11.7 TVA Bull Run Fossil Plant – Controller Cards 
1111 
11.8 Repairable Systems Reliability Growth Assessment 
1113 
11.9 Front Jounce Bumpers 
1114 
11.10 Transfer Case Seal 
1115 
11.11 Dental Acrylic Adhesive Fatigue 
1116 
11.12 DuaneCrowAMSAA Reliability Modeling 
1117 
11.13 Weibull Analysis Of Boiler Tube Failures 
1120 
11.14 Gas Turbine Seal Failures – A Batch Problem 
1123 
11.15 Challenger Space Shuttle Weibull 
1125 
Appendix A: Glossary 
A1 
Appendix B: Rank Regression And Correlation Method Of Weibull Analysis 
B1 
B1 Method 
B1 
B2 Example And StepByStep Procedure 
B1 
Appendix C: Maximum Likelihood Estimation* 
C1 
C1 Foreword 
C1 
C2 Statistics, Probability And Likelihood 
C1 
C3 The Likelihood Function 
C1 
C4 Maximizing The Likelihood Function 
C2 
C.5 Maximum Likelihood Example 
C3 
C.6 Interval MLE 
C6 
C.7 Maximum Likelihood Versus Median Rank Regression Estimates 
C8 
Appendix D: Goodness of Fit 
D1 
Appendix E: Weibayes Analysis 
E1 
E.1 Foreword 
E1 
E.2 Weibayes Equation With No Failures 
E1 
E.3 Weibayes With Failures 
E2 
Appendix F: Batch Failures Using The Aggregated Cumulated Hazard Function 
F1 
F.1 Batch Failures On Weibull Plots 
F1 
F.2 Batch Problems With The “PresentRisk” Method 
F2 
F.3 The ACH Method 
F3 
F.4 A Case Study: AeroEngines – (Lp Turbine Strap Failures) 
F4 
F.5 Concluding Remarks 
F5 
Appendix G: Weibull And Log Normal Mean And Variance 
G1 
G.1 Rth Moments 
G1 
G.2 Weibull Mean 
G2 
G.3 Weibull Variance 
G3 
G.4 Weibull Mode 
G3 
G.5 Weibull Median 
G3 
G.6 Log Normal Mean And Standard Deviation 
G3 
G.7 Log Normal Variance 
G4 
Appendix H: Weibull Graph Paper For Manual Plots 
H1 
Appendix I: Median Ranks 
I1 
Appendix J – Mixtures Of Populations And Failure Modes 
J1 
J.1 Competing Risk: 
J1 
J.3 Competing Risk Mixture: 
J2 
J.4 Compound Competing Risk Mixture: 
J2 
J.5 Weibath Model: 
J2 
J.7 Curve Shape. 
J3 
Appendix K: Answers To Problems 
K1 
Appendix L: The C4 Factor 
L1 
Appendix M: Graphical Repair Analysis 
M1 
Appendix N: Waloddi Weibull 
N1 
References 
R1 
Index 
I1 